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#1 21-09-2014 09:28:50
- mona123
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problem in topologie
Give an exemple of a coutable sequence of decreasing nonempty closed sets in Rn,whose intersection is empty
#2 21-09-2014 10:40:33
Re : problem in topologie
Hello,
[tex] A_n := \{ x=(x_1, x_2, \dots , x_n) \in \mathbb{R}^n, \, \forall \, 1\leq i \leq n, \, x_i \geq n \} [/tex].
EDIT : There's no contradiction between this exemple and this theorem : http://en.wikipedia.org/wiki/Cantor's_intersection_theorem , as [tex]A_n[/tex]'s diameter doesn't tend to 0.
Dernière modification par Choukos (21-09-2014 10:50:44)
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#3 21-09-2014 17:29:34
- mona
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Re : problem in topologie
can you pease tell me An 's diameter=?
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#5 21-09-2014 20:35:29
- mona
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Re : problem in topologie
in Rn diameter An=? in you example
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#6 21-09-2014 21:35:51
Re : problem in topologie
Hello again,
I've understood you correctly, however, I don't want to give the answer right away because I truly believe that you can answer it on your own, should you see what [tex]A_n[/tex] is in [tex]\mathbb{R}^2[/tex]. If you don't then I'll answer, but please, do try !
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#7 22-09-2014 01:38:26
- MOHAMED_AIT_LH
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Re : problem in topologie
Salut, Hello
[tex]F_k=[k,+\infty[[/tex] then[tex] (F_k)_{ k \geq 0}[/tex] sequence of decreasing nonempty closed sets of [tex]\mathbb R[/tex], but [tex]\cap F_k =\emptyset[/tex].
In general case [tex]{\mathbb R}^n[/tex], we can consider, for all [tex]k \in {\mathbb N}[/tex] the set : [tex]W_k=f^{-1}(F_k)[/tex] where [tex]f: {\mathbb R}^n \to \mathbb R[/tex] such that [tex]f(x)= \|x\|, \forall x \in {\mathbb R}^n.[/tex]
Since [tex]f[/tex] is continuouse and [tex]F_k[/tex] is a closed set of [tex]\mathbb R[/tex] , the set [tex]W_k[/tex] is a closed set of [tex]{\mathbb R}^n[/tex]. Since [tex]F_{k+1} \subset F_k[/tex], we have ; [tex]W_{k+1}=f^{-1}(F_{k+1}) \subset f^{-1}(F_k)=W_k[/tex] for all [tex]k \in {\mathbb N }[/tex]. Finally, we have: [tex]\cap W_k = \cap f^{-1}(F_k) = f^{-1} (\cap F_k)= \emptyset.[/tex]
Dernière modification par MOHAMED_AIT_LH (22-09-2014 04:01:12)
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